A numerical study of the Dufour and Soret effects on unsteady natural convection flow past an isothermal vertical cylinder

The Dufour and Soret effects on the unsteady laminar free convective flow with mass transfer flow past a semi-infinite isothermal vertical cylinder were studied numerically. The governing partial differential equations were converted into a non-dimensional form and solved numerically by applying a Crank-Nicolson type of implicit finite-difference method with a tri-diagonal matrix manipulation and an iterative procedure. For the hydrogen-air mixture, which is a non-chemical reacting fluid, the profiles of the unsteady dimensionless velocity, temperature and concentration are shown graphically for the different values of thermal and mass Grashof numbers, thermal diffusion parameters (Soret numbers) and diffusion-thermo parameters (Dufour numbers). Finally, the simulated values of the average skin-friction coefficient, the average Nusselt number and the average Sherwood number are presented. The numerical results reveal that for an increasing Soret number or decreasing Dufour number, the time to reach the temporal maximum and the steady-state decreases for the flow variables. As the Soret number increases or the Dufour number decreases, both the skin friction and the Sherwood number increase, whereas the Nusselt number decreases.     

Onset of buoyancy-driven convection in melting from below

When a horizontal homogeneous solid is melted from below, convection can be induced in a thermally unstable melt layer. In this study the onset of buoyancy-driven convection during time-dependent melting is investigated by using similarly transformed disturbance equations. The critical Rayleigh numbers based on the melt-layer thickness are found numerically for various conditions. For small superheats, the present predictions approach the well known results of classical Rayleigh-Bénard problems, that is, critical Rayleigh numbers are located between 1,296 and 1,708, regardless of the Prandtl number. However, for high superheats the critical Rayleigh number increases with an increase in phase change rate but with decrease in Prandtl number.     

Onset of convection in a porous mush during binary solidification

The onset of convection in a mushy layer during solidification of a binary melt is investigated by using the propagation theory we have developed. The critical conditions for the mushy-layer-mode of instability are obtained numerically for aqueous ammonium chloride solution. The mush thickness at the onset of convection is predicted as a function of the solution viscosity and compared with the experimental data. The onset time of the mushy-layer-mode convection has a minimum point with varying the superheat. When the superheat is large, the mushy layer grows slowly and a long time is required for the onset of convection.     

Onset of buoyancy-driven convection in isotropic porous media heated from below

A theoretical analysis of buoyancy-driven instability under transient basic fields is conducted in an initially quiescent, fluid-saturated, horizontal, isotropic porous layer. Darcy’s law is employed to explain characteristics of fluid motion, and Boussinesq approximation is used to consider the density variation. Under the principle of exchange of stabilities, a stability analysis is conducted based on the linear stability analysis and energy method and their modifications. The critical condition of onset of buoyancy-driven convection is obtained as a function of the Darcy-Rayleigh number. The propagation theory and the modified energy method under the self-similar coordinate suggest reasonable stability criteria and support each other. The former one based on the linear stability theory predicts more stable results than the latter based on the energy method. The growth period for disturbances to grow seems to be required until the instabilities are detected experimentally.     

The onset of double-diffusive convection in a stably stratified fluid layer heated from below

The time of the onset of double-diffusive convection in time-dependent, nonlinear temperature fields is investigated theoretically. The initially quiescent horizontal fluid layer with a uniform solute gradient experiences ramp heating from below, but its stable solute concentration is to reduce thermal effects which invoke convective motion. The related stability analysis is conducted on the basis of the propagation theory. Under the linear stability theory the thermal penetration depth is used as a length scaling factor and the linearized perturbation equations of similarity transform are solved numerically. The resulting correlations of the critical time to mark the onset of regular cells are derived as a function of the thermal Rayleigh and the solute Rayleigh numbers. The predicted stability criteria are apparently consistent with existing experimental results for aqueous solution of sodium chloride.     

Onset of solutal convection in liquid phase epitaxy system

The onset of convective instability in the liquid phase epitaxy system is analyzed with linear stability theory. New stability equations are derived under the propagation theory, and the dimensionless critical time τ _c to mark the onset of the buoyancy-driven convection is obtained numerically. It is here found that the critical Rayleigh number Ra _c is 8000, below which the flow is unconditionally stable. For Ra>Ra _c the dimensionless critical time τ _c to mark the onset of a fastest growing instability is presented as a function of the Rayleigh number and the Schmidt number. Available numerical simulation results and theoretical predictions show that the manifest convection occurs starting from a certain time τ _o (> τ _c ). It seems that during τ _c ≤τ≤τ _o secondary motion is relatively very weak. This article is dedicated to Professor Chang Kyun Choi for celebrating his retirement from the School of Chemical and Biological Engineering, Seoul National University.     

Onset of buoyancy-driven convection in the horizontal fluid layer heated from below with time-dependent manner

The onset of buoyancy-driven convection in an initially isothermal, quiescent fluid layer heated from below with time-dependent manner is analyzed by using propagation theory. Here the dimensionless critical time Τ_c to mark the onset of convective instability is presented as a function of the Rayleigh number Ra_Ø and the Prandtl number Pr. The present stability analysis predicts that Τ_c decreases with increasing Pr for a given Ra_Ø. The present predictions compare reasonably well with existing experimental results. It is found that in deep-pool systems the deviation of temperature profiles from conduction state occurs starting from a certain time Τ?(2～4) Τ_c .     

Free convection in power-law fluids from a heated sphere

In this work, the governing field equations describing heat transfer from a heated sphere immersed in quiescent power-law fluids have been solved numerically. In particular, consideration has been given to elucidate the role of Grashof number (Gr), Prandtl number (Pr) and power-law index (n), on the value of the Nusselt number (Nu) for a sphere in the natural convection regime. Further insights are provided by presenting streamline and constant temperature contours. The results presented herein encompass the following ranges of conditions: 10≤Gr≤10⁷; 0.72≤Pr≤100 and 0.4≤n≤1.8 thereby covering both shear-thinning and shear-thickening types of fluid behaviours. Broadly, all else being equal, shear-thinning behaviour can enhance the rate of heat transfer by up to three-fold where as shear-thickening can impede it up to ～30?40% with reference to that in Newtonian fluids. The paper is concluded by presenting detailed comparisons with the scant experimental data and the other approximate treatments of this problem available in the literature.     

Onset of buoyancy-driven instability in a pure melt solidified from above

When a pure liquid is solidified from above, convection may be induced in a thermally-unstable layer. The onset of buoyancy-driven convection during time-dependent solidification is investigated by using similarly transformed disturbance equations. The thermal disturbance distribution of the solid phase is approximated by WKB method, and the effects of various parameters on the stability condition of melt phase are analyzed theoretically. The present constant temperature cooling model gives more unstable results than the constant solidification velocity model of Smith [1].     

The temporal evolution of thermal instability in fluid layers isothermally heated from below

The temporal development of thermal disturbances in the fluid layer heated isothermally from below is investigated, based on propagation theory. This theory is examined by using scaling. To examine the behavior of thermal instability the mean-field approximation is employed and resulting equations are solved by Galerkin method. The stability criteria to mark the onset of convective instability are newly suggested as the intersection point of the growth rate of averaged temperature with that of its fluctuation. The resulting critical time is close to that derived from propagation theory. By considering the nonlinear effects, the characteristic times to represent the detection time of manifest convection and also to exhibit the minimum Nusselt number are discussed.     

Natural convection of nanofluids in a shallow rectangular enclosure heated from the side

This article reports an analytical and numerical study of natural convection in a shallow rectangular cavity filled with nanofluids. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure while the two horizontal ones are assumed insulated. Three types of nanoparticles are taken into consideration: Cu, Al₂O₃ and TiO₂. Various models are used for calculating the effective viscosity and thermal conductivity of nanofluids. The governing parameters for the problem are the thermal Rayleigh number, Ra, the Prandtl number Pr, the aspect ratio of the cavity, A and the solid volume fraction of nanoparticles, Φ. In the first part of the analytical study a scale analysis of the present problem is made. In the second part analytical solutions are obtained, in the limit of an infinite layer ($A\gg 1$ ), for the stream function and temperature fields. The analytical model relies on the assumption of a parallel flow approximation in the core region of the cavity and an integral form of the energy equation in the end regions. In the high Rayleigh number regime a good agreement is obtained between the predictions of the scale analysis and those of the analytical solution. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations based on the lattice Boltzmann method. Cet article présente une étude analytique et numérique de la convection naturelle dans une cavité rectangulaire peu profonde remplie de nanofluides. Les conditions aux limites de Neumann pour la température sont appliquées sur les parois verticales de l'enceinte tandis que les deux parois horizontales sont supposées isolées. Trois types de nanoparticules sont pris en considération: Cu, Al₂O₃ et TiO₂. Divers modèles sont utilisés pour le calcul de la viscosité et la conductivité thermique effective du nanofluide. Les paramètres régissant le problème sont le nombre de Rayleigh thermique, Ra, le nombre de Prandtl Pr, le facteur de forme de la cavité, A et la fraction volumique de nanoparticules solides, phi. Dans la première partie de l'étude analytique une analyse d'échelle du problème est réalisée. Dans la deuxième partie les solutions analytiques sont obtenues, dans la limite d'une couche infinie, pour la fonction de courant et les champs de température. Le modèle de la solution repose sur l'hypothèse de l'écoulement parallèle dans la région centrale de la cavité et sur un bilan de conservation d'énergie dans les extrémités. Pour des nombres de Rayleigh assez élevés un bon accord est obtenu entre les prédictions de l'analyse d'échelle et ceux de la solution analytique. Des solutions numériques des équations complètes sont obtenues pour une large gamme des paramètres de contrôle. Un bon accord est observé entre le modèle analytique et les simulations numériques basées sur la méthode de Lattice Boltzmann. © 2011 Canadian Society for Chemical Engineering     

Modeling non-Newtonian slurry convection in a vertical fracture

A model of non-Newtonian slurry convection in a fracture was developed. Based on the simulation [Eskin, D., Miller, M., 2008. A model of non-Newtonian slurry flow in a fracture. Powder Technol. 182, 313-322] and experimental [Tehrani, M.A., 1996. An experimental study of particle migration in pipe flow of viscoelastic fluids. J. Rheology 40, 1057-1077] results on particle migration across a fracture, an accepted modeling system is a three-layer flow consisting of the central core of high particle concentration surrounded by pure fluid layers. The obtained solution describes convection in a small fracture domain where both the mean shear rate and the local particle concentration are known. Numerical study of the developed model shows that the solids settling rate caused by convection is much higher (regularly, by a factor of 10-30) than the particle settling rate, calculated based on an assumption that the particle concentration is uniformly distributed across a fracture. The convection model can be incorporated in one of the known numerical codes for computation of slurry dynamics in a whole fracture. An engineering modification of the convection model allows computing particle slug transport in a fracture.     

Stability of buoyancy-driven convection in directional solidification of a binary melt

In directional solidification, compositional convection can be driven by an unstable density gradient in the melt. In this paper convective instabilities in liquid and mushy layers during solidification of a horizontal binary melt are analyzed by using the propagation theory. The self-similar stability equations are used to examine the boundary-mode and mushy-layer-mode of convection. The effects of velocity conditions at the liquid-mush interface on the onset of convection are discussed. The critical Darcy-Rayleigh number for the convection in the mushy layer decreases with increasing the temperature of a cooled boundary. This paper is presented on the occasion of professor Chang Kyun Choi’s retirement from the school of chemical and biological engineering of Seoul National University.     

Fully developed mixed convection of a binary fluid in a vertical porous channel

This paper reports an analytical and numerical study of mixed convection heat and mass transfer of a binary fluid in a vertical parallel plate channel filled with a porous medium. The thermal conditions applied on the walls of the system are uniform heat fluxes. Both the cases of double‐diffusion and Soret‐induced convection are considered. The governing equations for the porous medium rely on Darcy's model. The governing parameters for the problem are the Rayleigh number, Ra, Peclet number, Pe, Lewis number, Le, buoyancy ratio, φ, aspect ratio of the channel $A = L'/H'$ and the constant a (a = 0 for double diffusive convection and a = 1 for Soret induced convection). The resulting problem, in the limit of fully developed mixed convection, is solved analytically in closed form. A numerical solution of the full governing equations is demonstrated to be in good agreement with the analytical model. The temperature and velocity fields and the Nusselt and Sherwood numbers are obtained in terms of the governing parameters. The possible existence of reversed flows in the channel is discussed. © 2011 Canadian Society for Chemical Engineering     

Prediction of melting process driven by conduction-convection in a cavity heated from the side

A fixed-grid finite volume numerical approach is developed to simulate the melting during the solid-liquid phase-change driven by convection as well as by conduction. This approach adopts the enthalpy-porosity method augmented with the front-layer predictor-corrector and the pseudo Newton-Raphson algorithms that were devised to track the phase front efficiently in the conduction-driven phase-change problems. The computational results compare well with experimental data and transformed-grid results in the literature. Also, the effect of the delayed heat-up at a heated wall on the melting process is investigated.     

Oscillatory convection in a horizontal porous layer saturated with a viscoelastic fluid

When a horizontal porous layer saturated with a viscoelastic liquid is heated from below, the onset conditions of thermal convection are found to be functions of Darcy-Rayleigh number, wave number, and viscoelastic properties. In this study, the linear stability was studied analytically in order to investigate the viscoelastic effects of saturated liquids on the onset conditions in connection with oscillatory instabilities at the threshold of stationary convection. It is suggested that the resulting oscillatory instabilities occur at lower values of Darcy-Rayleigh number than the critical value for the stationary convection. From the occurrence of oscillatory instabilities of viscoelastic liquid, it is expected that the periodic motion should be replaced by stationary modes in a horizontal porous layer.     

Natural convection heat transfer of nanofluids along a vertical plate embedded in porous medium

The unsteady natural convection heat transfer of nanofluid along a vertical plate embedded in porous medium is investigated. The Darcy-Forchheimer model is used to formulate the problem. Thermal conductivity and viscosity models based on a wide range of experimental data of nanofluids and incorporating the velocity-slip effect of the nanoparticle with respect to the base fluid, i.e., Brownian diffusion is used. The effective thermal conductivity of nanofluid in porous media is calculated using copper powder as porous media. The nonlinear governing equations are solved using an unconditionally stable implicit finite difference scheme. In this study, six different types of nanofluids have been compared with respect to the heat transfer enhancement, and the effects of particle concentration, particle size, temperature of the plate, and porosity of the medium on the heat transfer enhancement and skin friction coefficient have been studied in detail. It is found that heat transfer rate increases with the increase in particle concentration up to an optimal level, but on the further increase in particle concentration, the heat transfer rate decreases. For a particular value of particle concentration, small-sized particles enhance the heat transfer rates. On the other hand, skin friction coefficients always increase with the increase in particle concentration and decrease in nanoparticle size.     

The role of convection and turbulent dispersion in blending

Blending is an important operation in process industries. Under turbulent conditions, it occurs via convection and turbulent dispersion. In the present work, attempts have been made to identify the controlling mechanism, by comparing the blending performance of three different equipments: stirred tanks, jet mixers and ultrasound mixers. It is seen that jet mixers offer lower mixing time as compared with impeller mixers, whereas ultrasonic horns show higher mixing time as compared with impeller mixers, when compared at the same level of power consumption in tanks having the same size.